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Estimating principal curvatures and principal directions of a smooth surface represented by a triangular mesh is an important step in many CAD or graphics related tasks. This paper presents a new method for curvature tensor estimation on a triangular mesh by replacing flat triangles with triangular parametric patches. An improved local interpolation scheme of cubic triangular Bezier patches to vertices and vertex normals of triangle meshes is developed. Piecewise parametric surfaces that have C^0 continuity across boundary curves of adjacent patches and G^1 continuity at the joint vertices are obtained by the interpolation scheme. A closed form expression of Taubin integral-a 3x3 symmetric matrix in integral formulation-is derived based on the piecewise parametric surfaces. Principal curvatures and principal directions are then computed from the Taubin integral. The proposed method does not need to parameterize data points or solve a linear system which is usually required by other surface fitting methods. Compared to several state-of-the-art curvature estimation methods, the proposed method can generate more accurate results for general surface meshes. The experiments have demonstrated its accuracy, robustness and effectiveness.